Admissible subcategories of derived categories of noncommutative curves

Abstract

We study admissible subcategories of the derived categories of smooth noncommutative (nc) curves as classified by Reiten–van den Bergh. We prove that any admissible subcategory of the derived category of a smooth nc curve is again the derived category of a smooth nc curve. We use this result to classify semiorthogonal decompositions in derived categories of nc curves. The results obtained imply that phantom categories do not exist in these cases. As a further application, we prove an extension of the Bondal–Orlov reconstruction theorem to the case of orbifold curves.